Entangling polaritons via dynamical Casimir effect in circuit quantum electrodynamics

We investigate theoretically how the dynamical Casimir effect can entangle quantum systems in different coupling regimes of circuit quantum electrodynamics, and show the robustness of such entanglement generation against dissipative effects, considering experimental parameters of current technology. We consider two qubit-resonator systems, which are coupled by a SQUID driven with an external magnetic field, and explore the entire range of coupling regimes between each qubit and its resonator. In this scheme, we derive a semianalytic explanation for the entanglement generation between both superconducting qubits when they are coupled to their resonators in the strong coupling regime. For the ultrastrong and deep strong coupling regimes, we design experimentally feasible theoretical protocols to generate maximally entangled polaritonic states.

Figure 1

Pictorial illustration of the system from a quantum-optical perspective. Two single-mode cavities, each one containing a qubit, are sharing a partially reflecting mirror. When such a mirror moves relativistically, the generation of DCE photons into the cavities can highly entangle the two subsystems.

Figure 2

(a) Time evolution of concurrence between the two qubits (solid line) and mean photon number of the first resonator (dashed-dotted line). Each qubit is resonantly coupled to its corresponding resonator with the same interaction strength $g_0=g_1=g_2=0.04{\omega }_1$, with ${\omega }_2=1.25{\omega }_1$ and ${\alpha }_0=0.1g_0$. The vertical dotted line represents the time ${\omega }_1{t}_{\text{SO}}=1200$ when the SQUID external flux is switched off. The dashed line is the concurrence computed using Eq. (8). (b) Real and imaginary parts of the reduced density matrix of the two-qubit system at $t_{\text{SO}}$. The semianalytical model developed in this work agrees with our numerical studies, some of which were previously presented in Ref. [9].

Figure 3

Energy-level diagram of the Hamiltonian given by Eq. (5), which consists in a harmonic ladder and an anhamonic part. The dashed arrows stand for the transitions induced by the two-mode squeezing term, where only the transitions between adjacent states of the harmonic ladder are induced resonantly.

Figure 4

Time evolution of (a) the concurrence and (b) the mean photon number when there is dissipation (solid lines) in comparison to the case when there is not (dashed lines). Here, we consider ${\kappa }_{\mu }={\gamma }_{\mu }={\gamma }_{\mu }^{\text{ph}}=5\times {10}^{-6}{\omega }_1$ and the same parameters of Fig. 2, except $t_{\text{SO}}$ and ${\alpha }_0=0.2g_0$. The shaded area stands for the numerical errors because of the truncation of the Fock basis describing the resonators.

Figure 5

Energy spectrum of the quantum Rabi model with respect to the ratio $g_0/\omega$, where $\omega$ is the frequency of both resonator mode and qubit, and $g_0$ is the coupling strength of the qubit-resonator interaction. The dashed-dotted and solid lines stand for the eigenstates with positive and negative parity, respectively. The arrows indicate the energy gap between the ground and first excited states for $g_0/\omega =(0.25,0.5,0.75)$. If we resonantly induce a transition from the ground to the first excited state by a single-photon process, the probability of inducing a transition from the first excited state to the next allowed one (next dashed-dotted line) by the same process is very small due to nonresonance. Then, the strong anharmonicity of the energy spectrum in the USC regime ($0.1\lesssim g_0/\omega <1$) allows us to work in a low-energy subspace, that is, the ground and the first excited states of a quantum Rabi system in the USC regime can be used as a qubit [36].

Figure 6

(a) Time evolution of the concurrence (solid line) between two identical quantum Rabi systems computed using the Hamiltonian without approximations, Eq. (10), considering $g_0=0.15\omega$ and ${\alpha }_0=0.05g_0$. The circles represent the concurrence computed via Eq. (11), $\mathcal{C}=|sin(\lambda t\left)\right|$, which is in good agreement with the exact numerical solution. Let $P\left(\right|{\phi }_k^{\mu }\rangle )$ the probability of the subsystem $\mu$ being in the eigenstate $|{\phi }_k^{\mu }\rangle$, the dashed line stands for $P\left(\right|{\phi }_0^1\rangle )+P\left(\right|{\phi }_1^1\rangle )$, which shows us the validity of the assumption that each quantum Rabi subsystem in the USC regime works as a qubit since we have $P\left(\right|{\phi }_0^1\rangle )+P\left(\right|{\phi }_1^1\rangle )\sim 1$ during the evolution. (b) Similar to (a) in the case when dissipative processes take place. The solid line and the circles refer to the solution of Eqs. (13) and (15, 16, 17, 18, 19), respectively, considering $\kappa =\gamma =0.5\times {10}^{-4}\omega$ and $t_{\text{SO}}=\pi /2\lambda$ (dotted line).

Figure 7

Von Neumann entropy of the first hybrid subsystem calculated via Eq. (1) (solid line) and Eq. (27) (dashed line) using ${\omega }_2=1.25{\omega }_1,\mathrm{\Omega }={\omega }_1,\beta =1.5$, and ${\alpha }_0=0.01{\omega }_1$ (upper panel), and ${\omega }_2=1.25{\omega }_1,\mathrm{\Omega }=0.1{\omega }_1,\beta =1.5$, and ${\alpha }_0=0.005{\omega }_1$ (lower panel).